300 Lagrange’s formula

worked with the Paris Académie to standardize the

weights and measures of the day. The committee for-

mulated the metric system and advocated the general

use of a decimal base.

Lagrange’s significant achievements were recog-

nized by the emperor Napoleon in 1808, when he was

named to the Legion of Honour and a count of the

empire. Five years later he was also named Grand

Croix of the Ordre Impérial de la Réunion. He died in

Paris, France, on April 10, 1813.

Lagrange’s impact in mathematics, especially in

mathematical physics, is still felt today. Many funda-

mental concepts in mechanics and multivariable calcu-

lus—such as the Lagrangian (the difference between

kinetic energy and potential energy of a set of parti-

cles), the Lagrangian description (a measure of defor-

mation of a physical body), and Lagrange multipliers

in calculus—play a vital role in the current study of

these subjects.

See also L

AGRANGE

’

S FORMULA

.

Lagrange’s formula (Lagrange’s interpolation formula)

Given a collection of points on the plane, it is sometimes

desired to find the formula for a function that passes

through each of those points. For example, a scientist

may seek a formula for a function that fits all the data

values obtained from an experiment. In the late 1700s,

Italian-French mathematician J

OSEPH

-L

OUIS

L

AGRANGE

suggested the following

INTERPOLATION

formula:

If (a1,b1), (a2,b2), …, (an,bn) are a collection

of points in the plane, with the values aidis-

tinct, then

is a

POLYNOMIAL

, of degree n– 1, that passes

through each of the points.

This formula is today known as Lagrange’s formula.

One can check that it works by substituting x= a1to

see that f(a1) = b1, and so on.

As an example, consider the points (1,2), (2,5), and

(3,1) in the plane. Lagrange’s formula shows that the

quadratic

passes through each of them.

Many intelligence tests ask participants to identify

“the next number in the sequence.” Lagrange’s for-

mula provides a means for justifying absolutely any

answer to such a question. For example, the next

number in the sequence 2,4,6,… could well be 103.

We can argue that the sequence follows the formula:

. (Apply Lagrange’s

formula to the points (1,2), (2,4), (3,6), and (4,103).)

Lambert, Johann Heinrich (1728–1777) Swiss-

German Geometry, Analysis, Number theory, Physics

Born on August 26, 1728, scholar Johann Lambert is

best remembered as the first to prove, in 1761, that πis

an

IRRATIONAL NUMBER

. He also worked on Euclid’s

PARALLEL POSTULATE

and came close to the discovery of

NON

-E

UCLIDEAN GEOMETRY

. Lambert also developed

the notation and the theory of

HYPERBOLIC FUNCTIONS

.

In 1766 Lambert wrote Theorie der Parellellinien

(On the theory of parallel lines), in which he postulated

the existence of surfaces on which triangles have angu-

lar sums less than 180°, thereby yielding an example of

a geometry in which the parallel postulate would be

false. (Such a surface was later discovered. It is called a

pseudosphere.) Lambert proved that in this geometry,

the sum of the angles of a triangle would not be con-

stant, and in fact would increase (but never to equal

180°) as its area decreases.

In 1737 L

EONHARD

E

ULER

had proved that eand

e2are both irrational. In the paper “Mémoire sur

quelques propriétés remarquables des quantités tran-

scendantes circulaires et logarithmiques” (Memoir on

some remarkable properties of transcendental quanti-

ties circular and logarithmic) presented to the Berlin

Academy of Sciences in 1761, Lambert provided a

proof that if xis a rational number different from zero,

then neither exnor tan xcan be rational. Thus Lambert